Note that an upper bound of the number of adjacent task swappings needed from \(\Gamma _{\pi _1}^{R(n)}\)  to reach \(\Gamma _{\pi _2}^{R(n)}\)  is subject to the diameter of the modified bubble-sort graph \(MBS_n\)  discussed in Section . To the best of our knowledge, the formula of the diameter of \(MBS_n\)  is not known [1]}, [2]}, [3]}. Nevertheless, a minimum-length sequence of sorting an aribtrary circular permutation \(p \in \mathfrak {S}_n\)  into the identity permutation \(I\)  using the generating set \(S_6=\lbrace (i\,i+1):1 \le i < n\rbrace  \cup \lbrace (1\,n)\rbrace \)  in Algorithm  is obtained in polynomial time [4]}. It follows that Algorithm  runs in polynomial time as with other algorithms involving permutation sortings discussed in this paper.
